arXiv subjects

Jan de Gier

Publications and source records attributed to Jan de Gier.

66 records · Page 4Linked to original sources

arXiv2002

Stochastic processes and conformal invariance

We discuss a one-dimensional model of a fluctuating interface with a dynamic exponent $z=1$. The events that occur are adsorption, which is local, and desorption which is non-local and may take place over regions of the order of the system size. In the thermodynamic limit, the time dependence of the system is given by characters of the $c=0$ conformal field theory of percolation. This implies in a rigorous way a connection between CFT and stochastic processes. The finite-size scaling behavior of the average height, interface width and other observables are obtained. The avalanches produced during desorption are analyzed and we show that the probability distribution of the avalanche sizes obeys finite-size scaling with new critical exponents.

cond-mat.stat-mech↗

arXiv2002

Temperley-Lieb Stochastic Processes

We discuss one-dimensional stochastic processes defined through the Temperley-Lieb algebra related to the Q=1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics.

math-ph↗

arXiv2001

The XXZ spin chain at $Δ=- {1/2}$: Bethe roots, symmetric functions and determinants

A number of conjectures have been given recently concerning the connection between the antiferromagnetic XXZ spin chain at $Δ= - \frac12$ and various symmetry classes of alternating sign matrices. Here we use the integrability of the XXZ chain to gain further insight into these developments. In doing so we obtain a number of new results using Baxter's $Q$ function for the XXZ chain for periodic, twisted and open boundary conditions. These include expressions for the elementary symmetric functions evaluated at the groundstate solution of the Bethe roots. In this approach Schur functions play a central role and enable us to derive determinant expressions which appear in certain natural double products over the Bethe roots. When evaluated these give rise to the numbers counting different symmetry classes of alternating sign matrices.

math-ph↗

arXiv2001

Critical Q=1 Potts Model and Temperley-Lieb Stochastic Processes

We consider the groundstate wave function and spectra of the $L$-site XXZ $U_q[s\ell(2)]$ invariant quantum spin chain with $q=\exp(πi/3)$. This chain is related to the critical Q=1 Potts model and exhibits $c=0$ conformal invariance. We show that the problem is related to Hamiltonians describing one-dimensional stochastic processes defined on a Temperley-Lieb algebra. The bra groundstate wave function is trivial and the ket groundstate wave function gives the probabilty distribution of the stationary state. The stochastic processes can be understood as interface RSOS growth models with nonlocal rates. Allowing defects which can hop on the interface one obtains stochastic models having the same stationary state and spectra (but not degeracies) as the XXZ chain.

cond-mat.stat-mech↗

arXiv2001

Exact stationary state for a deterministic high speed traffic model with open boundaries

An exact solution for a high speed deterministic traffic model with open boundaries and synchronous update rule is presented. Because of the strong correlations in the model, the qualitative structure of the stationary state can be described for general values of the maximum speed. It is shown in the case of $v_{\rm max}=2$ that a detailed analysis of this structure leads to an exact solution. Explicit expressions for the stationary state probabilities are given in terms of products of $24\times 24$ matrices. From this solution an exact expression for the correlation length is derived.

nlin.CG↗

arXiv1998

Exact Stationary State for an ASEP with Fully Parallel Dynamics

The exact stationary state of an asymmetric exclusion process with fully parallel dynamics is obtained using the matrix product Ansatz. We give a simple derivation for the deterministic case by a physical interpretation of the dimension of the matrices. We prove the stationarity via a cancellation mechanism and by making use of an explicit representation of the matrix algebra we easily find closed expressions for the correlation functions in the general probabalistic case. Asymptotic expressions, obtained by making use of earlier results, allow us to derive the exact phase diagram.

cond-mat.stat-mech↗

arXiv1997

Bethe Ansatz solution of a decagonal rectangle triangle random tiling

A random tiling of rectangles and triangles displaying a decagonal phase is solved by Bethe Ansatz. Analogously to the solutions of the dodecagonal square triangle and the octagonal rectangle triangle tiling an exact expression for the maximum of the entropy is found.

cond-mat.stat-mech↗

arXiv1997

Solvable Rectangle Triangle Random Tilings

We show that a rectangle triangle random tiling with a tenfold symmetric phase is solvable by Bethe Ansatz. After the twelvefold square triangle and the eightfold rectangle triangle random tiling, this is the third example of a rectangle triangle tiling which is solvable. A Bethe Ansatz solution provides in principle an accurate estimate of the entropy and phason elastic constants. In the twelvefold and eightfold cases even exact analytic expressions have been obtained from the Bethe Ansatz solution.

cond-mat.stat-mech↗

arXiv1996

On the integrability of the square-triangle random tiling model

It is shown that the square-triangle random tiling model is equivalent to an asymmetric limit of the three colouring model on the honeycomb lattice. The latter model is known to be the O(n) model at T=0 and corresponds to the integrable model connected to the affine $A_2^{(1)}$ Lie algebra. Thus it is shown that the weights of the square-triangle random tiling satisfy the Yang-Baxter equation, albeit in a singular limit of a more general model. The three colouring model for general vertex weights is solved by algebraic Bethe Ansatz.

solv-int↗

arXiv1996

The exact solution of an octagonal rectangle triangle random tiling

We present a detailed calculation of the recently published exact solution of a random tiling model possessing an eight-fold symmetric phase. The solution is obtained using Bethe Ansatz and provides closed expressions for the entropy and phason elastic constants. Qualitatively, this model has the same features as the square-triangle random tiling model. We use the method of P. Kalugin, who solved the Bethe Ansatz equations for the square-triangle tiling, which were found by M. Widom.

solv-int↗

arXiv1996

Operator Spectrum and Exact Exponents of the Fully Packed Loop Model

We develop a Coulomb gas description of the critical fluctuations in the fully packed loop model on the honeycomb lattice. We identify the complete operator spectrum of this model in terms of electric and magnetic {\em vector}-charges, and we calculate the scaling dimensions of these operators exactly. We also study the geometrical properties of loops in this model, and we derive exact results for the fractal dimension and the loop size distribution function. A review of the many different representations of this model that have recently appeared in the literature, is given.

cond-mat↗

arXiv1996

Exact Solution of an Octagonal Random Tiling Model

We consider the two-dimensional random tiling model introduced by Cockayne, i.e. the ensemble of all possible coverings of the plane without gaps or overlaps with squares and various hexagons. At the appropriate relative densities the correlations have eight-fold rotational symmetry. We reformulate the model in terms of a random tiling ensemble with identical rectangles and isosceles triangles. The partition function of this model can be calculated by diagonalizing a transfer matrix using the Bethe Ansatz (BA). The BA equations can be solved providing {\em exact} values of the entropy and elastic constants.

solv-int↗